Aperture optimized RTM migration Matteo Giboli*, Cyril Agut, TOTAL SA and Reda Baina OPERA Summary Stacking is of paramount importance in seismic processing to improve the signal to-noise ratio (S/N) and the imaging quality of seismic data. The conventional stacking method that averages equally a collection of input traces cannot robustly suppress coherent noise. To attenuate this kind of noise and achieve an optimally stacked image remains an attractive and challenging research topic in the seismic industry. The key point for ongoing research is to develop methods that can be used to reliably discriminate between “good” and “bad” data samples. To this end it is important to identify two key objectives for the process of optimal stacking. The first is to find a suitable domain where signal can be easily distinguished from noise and the second is to build a robust procedure that allows only the signal contribute to the final migrated stack. We describe a novel, iterative method to clean and enhance the stacked migration image. For band-limited migration algorithms we define an original pre-stack image domain that is analogous to the aperture partitioned migration domain of Kirchhoff-type migration techniques. Through an automatic procedure that is based on a coherence analysis we show how, in this domain, signal can be separated from both coherent and incoherent noise in an effective way. With the aid of synthetic examples we show how this yields to a superior quality image compared to a conventional migration stack. Introduction Recent advances in seismic acquisition techniques have considerably improved the illumination of the subsurface. This capability to better compensate for uneven illumination represents a boon for production of new images of the subsurface. Seismic acquisitions are typically designed to be strongly redundant, in order to improve the signal to noise ratio; however, more data potentially means more diversity in the quality across different partitioned data. Wide azimuth acquisition may yield localized data areas that are “poor” in quality for certain azimuths, and will penalize the “good” data yielded in those locations on other azimuth cubes when processed into the final stack. Additionally, the inadequacy of the physics we use at various stages of the seismic processing workflow does not allow to fully benefit from all the new available measurements. Inaccuracy of the velocity models produces errors in the computed travel-times and these lead to inconsistencies in the positioning of seismic events in different pre-stack images; if not properly aligned before summation they can interfere destructively yielding to a defocused migrated image. Advances in velocity estimation

techniques will certainly lead to better subsurface models over time, but a certain amount of inaccuracy in our velocity model is something we will have to deal with for a long time. Recent years have seen a flurry of activity on various flavors of image enhancement through optimized summation of pre-stack migrated images. Most of them stem from the observation that conventional stacking works well only when all traces have similar amplitudes and S/N and when the noise patterns are statistically independent of the noise of any other trace and of the signal (Mayne, 1962; Robinson, 1970; Neelamani et al., 2006). The basic idea is to enhance the signal-to-noise ratio by only stacking those volumes that contain consistent and relevant information. Liu et al. (2009) and Compton et al. (2012) propose weighting traces according to a measure of the local correlation between each input trace and a conventional stacking reference trace. Kun et al. (2014) apply a template matching technique between pre-stack images and a windowed reference image to find an optimal alignment and the weighting coefficients. These procedures are semi-automatic and the quality of the final stack is to some extent dependent on the quality of the reference trace. Local correlation stacking methods can benefit from more advanced techniques to obtain the reference trace (Sanchis and Hanssen, 2011). In the following we present a method to sort pre-stack migrated images obtained with wave-equation migration algorithms into aperture indexed common image gathers (CIGs). In analogy to conventional Kirchhoff-type migration techniques this parameter represents the spatial region around the common midpoint over which a diffraction stack is carried out. Optimal stacking is achieved by means of an automatic procedure where no reference stack is required. Kirchoff type aperture for band-limited migration An optimal gather should have enough sensitivity to the velocity errors that can be accurately measured and effectively fed back to the image gather. The signal should show specific characteristics that can allow us to design a robust criterion to distinguish between useful information and, ideally, both coherent and incoherent noise. Lastly, this common image gather should be practical to compute. The post-migration offset-aperture domain uncovers interesting properties of the wavefield (Kabbej et al. 2005, Giboli et al. 2013). These partitioned images act as proxies for incident and scattered illumination, which in themselves

Aperture optimized RTM migration

are valuable for aiding interpretation, especially in subsalt regions with poor illumination. Migration aperture is a critical parameter in Kirchhoff migration to obtain the best image quality from a given dataset. Reducing the migration aperture generally enhances the signal/noise ratio but harms the imaging of dipping events. Unfortunately, wave-equation migration methods cannot directly follow an analogous methodology. Actually, there is no unequivocal definition of migration aperture for common-shot migration methods. It may represent, for each shot, the distance between the extrema of active receivers position and those of the computation grid but it is more often referred to the horizontal distance between the source and the image location (Figure 1).

.

Figure 1: horizontal distance between the source and the image location is usually referred to as “aperture” for RTM. There is, however (especially in moderately complex models) a strong relation between the directional angle gathers and the “conventional” aperture that is the

displacement vector between the source-receiver midpoint and the image location (and between the opening angles and the offset vector). For band-limited migration it is possible to obtain an analogous aperture parameter by means of the “double migration” method (Bleistein, 1987). This technique, originally developed for ray-theory can be successfully applied in case of band-limited propagation (Giboli et al. 2012). We proceed in the same way. In frequency domain we have the conventional migrated shot

𝐼(𝒙, 𝒔) = �𝑑𝜔 𝑢(𝒙, 𝒔,𝜔)𝑣∗(𝒙, 𝒔,𝜔),

and two attribute migrated images

𝐼ℎ𝑥(𝒙, 𝒔) = �𝑢(𝒙, 𝒔,𝜔)(𝑠𝑥 − 𝑟𝑥)𝑣∗(𝒙, 𝒔, 𝒓,𝜔) 𝑑𝒓 𝑑𝜔

and

𝐼ℎ𝑦(𝒙, 𝒔) = �𝑢(𝒙, 𝒔,𝜔)�𝑠𝑦 − 𝑟𝑦�𝑣∗(𝒙, 𝒔, 𝒓,𝜔) 𝑑𝒓 𝑑𝜔

we perform the following divisions

ℎ𝑥(𝒙, 𝒔) =𝐼ℎ𝑥(𝒙, 𝒔)𝐼(𝒙, 𝒔) and ℎ𝑦(𝒙, 𝒔) =

𝐼ℎ𝑦(𝒙, 𝒔)𝐼(𝒙, 𝒔) .

We then compute two new quantities

𝑎𝑥(𝒙, 𝒔) =ℎ𝑥(𝒙, 𝒔)

2 + 𝑠𝑥 − 𝑥 and

𝑎𝑦(𝒙, 𝒔) =ℎ𝑦(𝒙, 𝒔)

2 + 𝑠𝑦 − 𝑦. Finally, we use the obtained mapping 𝒂𝒙(𝒙, 𝒔), 𝒂𝒚(𝒙, 𝒔) and the conventional reflectivity 𝑰(𝒙, 𝒔) to build the pre-stack 5D image cube in the aperture domain. At each subsurface point, these image gathers become functions of only two aperture parameters, where integration over the two offset coordinates is performed implicitly. Actually, the mapping into a 7D offset-aperture domain would be meaningless. Apart from the massive amount of memory requirements throughout the mapping process and a huge amount of disk space to store the results this type of mapping comes after wave-field migration where integration over receiver position has already been carried out during propagation. We therefore propose splitting the general decomposition process into a complementary aperture-domain and offset-domain image gathers. We follow the derivation of our RTM aperture optimized imaging.

Aperture optimized RTM migration

Methodology For band-limited migrations only a portion of the migration operator is generally visible since the multi-fold stacking has already been carried out during back-propagation of the receiver wave-field. However because of inaccuracies in the velocity model, approximations of the physics underlying our propagation methods or irregular/insufficient sampling of input data, this weighted stacking of migration operators may be incomplete. At the imaging step this yields to migration noise and artefacts due to operator or data aliasing, truncated aperture or non-optimal migrated weights. In order to tackle different problems at separate stages of the procedure the proposed method consists of three key steps: • We compute surface offset gathers by means of attribute

migration. Residual move-out picking should then give us for each location an estimation of the error in the velocity and the range of most reliable offsets.

• We compute attribute aperture gathers taking care to apply a residual moveout correction and an offset mute using the quantities computed at the previous step.

• Finally optimal aperture selection is performed using the procedure described below.

We use a technique similar to the method developed in Giboli et al. (2012), where the S/N ratio of the image stack is improved by using an optimal selection of the migration operator which includes mostly the specular energy and not the long “tails” of the migration operator. The optimal stack 𝐼(𝒙) can be found as a two-step procedure where the pre-stack solution 𝑝� of the minimization problem min𝑝

‖𝑝(𝝁,𝜶,𝝆)‖1 𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜 𝐾𝑝(𝝁,𝜶,𝝆) = 𝐼(𝒙,𝒂)

is followed by a weighted stack

𝐼(𝒙) = �𝐾𝑤(𝝁,𝜶,𝝆)𝑝�(𝝁,𝜶,𝝆)𝒂

.

At each iteration, we efficiently solve the minimization problem by defining a least squares formulation using the transformation matrix 𝑈 = 𝐾−1 and using a matching pursuit scheme to promote sparseness in the model space. The transformed axis 𝝁,𝜶 denote the dual of axis 𝒙,𝒂 domain whereas 𝝆 represents the extra dimensions (i.e. wavenumber) where signal should be enough focused to allow for an easy discrimination from unwanted noise. We choose a local transform, which has the property of time-frequency and space-wavenumber localization. When the transform is done in the time-space domain, the aliased energy does not spread since the basis has local support.

The weights 𝑤 allow to keep only horizontal features along the aperture domain above a given coherence threshold and events whose measured dip fall within the equivalent aperture class. An ideal transform should provide multi-scale representation where reliable estimation of specular energy from low wavenumbers can be updated on the next scale during iterations. Synthetic Data Example We illustrate the proposed method using a section of the BP benchmark model (Billette and Brandsberg-Dahl, 2005). We demonstrate how errors in the velocity model (especially around the salt body) and irregularities in the acquisition that typically occur when dealing with real data-set would affect the image quality. We therefore use a slightly modified version of the original model as the migration velocity model and a realistic seismic acquisition profile obtained from an available 2D data-set. The difference between this modified model and the original model is shown in Figure 2.

Figure 2: BP model. Difference between the migration velocity model and the true velocity model. An RTM image using the modified velocity model and the true density model after standard post-processing is displayed in Figure 4. Because of the inadequacies in the velocity models the migrated stack is polluted by many artifacts coming from unfocused energy. Figure 5 shows the stack obtained with conventional RTM but using the true velocity model. It is possible to appreciate how inaccuracies in the velocity models and irregularity in the acquisition can be highly detrimental to RTM, especially in sub-salt areas where geology may be rather complex. Figure 3 compares aperture CIGs obtained from raw RTM migrated shots after residual move-out correction and offset mute (top) and optimized aperture CIGs (bottom). Application of our methodology to these CIGs yields to the final image (Figure 6). Yellow boxes highlight areas where

Aperture optimized RTM migration

migration noise is dramatically decreased allowing for an easier interpretation of seismic events. Yellow circles show a considerable improvement in the continuity of those horizons.

Figure 3: Raw RTM aperture CIGs (top) and aperture optimized RTM CIGs (bottom).

Figure 4: Conventional image obtained with wrong velocity model.

Figure 5: Conventional image obtained with true velocity model.

Figure 6: Optimized image obtained after post-processing of migrated aperture gathers with wrong model. Conclusion We propose a novel methodology for improving the S/N ratio of stacked images from band-limited migration. The key element is the decomposition of migrated shots into an original attribute aperture domain that allows detection of the specular energy of band-limited migration operators. The process is improved using local transform allowing multi-scale analysis. This technique attenuates aliasing and migration operator artifacts and improves continuity of seismic events.

Aperture optimized RTM migration

M. Giboli, C.Agut, R.Baina

We only sum in a migration window to

reduce noise save time

Aperture

Migration

cone

Source Mid-point Receiver

Trace

Event

Reflector

Migration

Operator

What is migration aperture?

We want to use the “right” migration window

Source Mid-point Receiver

Specular ray

Dipping reflector

Offset

Aperture

Dipping events imaging

Motivations

Optimal aperture is a critical parameter for best image quality.

• Signal/Noise ratio is improved.

• Bad/missing traces artifacts are eliminated.

• Aliasing noise as acquisition footprint is reduced.

The Fresnel zone summation

p

p

L

R S

R S

Fresnel aperture

L

Depth domain

Time domain

L

Optimal aperture is linked to the

Fresnel aperture in the data domain.

Choosing optimal aperture in the image

domain has two advantages:

1. Migration unfolds the complexity of

the reflected wave-field

2. No need to remigrate for

parameter tuning

The Fresnel zone: common-shot example

The Fresnel zone: common-shot example

The Fresnel zone: common-shot example

Common aperture domain CIGs

Fresnel zone

Artifacts

Aperture

Dep

th

RTM Aperture gather

Conventional RTM imaging condition

Attribute aperture migration

Conventional RTM imaging condition

migration attribute

Attribute aperture migration

The surface attribute h

Attribute aperture migration

The surface attribute h

The aperture attribute x*

Finally,

xsh

ss 2

),(),(* x

xx

dssxsxIxI )),((),(),( *xxx

Attribute aperture migration

Aperture gather

Aperture gathers for RTM

Aperture gather

Aperture gathers for RTM

Aperture gather

• Aperture binning can be computed

in the same way (and at the same

time) as for surface offset

• A mute law in the aperture domain

can be defined to filter out

unwanted features

CIG

S=R

Aperture bin

Dip angle bin

Aperture

CIG

S

Reflection angle bin

R

Offset bin

Common-offset, aperture, reflection and dip angle

Artifact

CIG Stack

image

Model

Coherent noise: look in different domains

Artifact

Offset

Artifact

CIG Stack

image

Model Common Offset

Coherent noise: look in different domains

Common Offset

Aperture

Artifact

Artifact

Common Aperture

Offset

Artifact

CIG Stack

image

Model

Coherent noise: look in different domains

Anti-aliasing

CIG

RTM Common

offset plane

RTM Common

aperture plane

Aliasing Aliasing

CIG

Dep

th

Aliasing

Aperture x Offset x

Dep

th

Anti-aliasing

Aliasing!

RTM Conventional stack RTM Optimized stack

x

Dep

th

Dep

th

x

Anti-aliasing

Aliasing

Offset x

Aliasing!

Anti-aliasing

Aperture x

• Cost is almost 0

Remarks

• Cost is almost 0

• 100 % of production studies using band-limited

migration

Remarks

• Cost is almost 0

• 100 % of production studies using band-limited

migration

• 5 dimensions cubes

Remarks

RTM/WEM

OFFSET Gathers

POST-PROC.

APERTURE Gathers

POST-PROC.

optimal aperture

RTM/WEM

APERTURE Gathers

POST-PROC.

OFFSET Gathers

POST-PROC.

(i.e. demul, RMO, Mute)

Specularity +

aperture mute law

RMO +

offset mute law

Possible workflows

-500 -300 -100 100 300 500

BP MODEL: MIGRATION VELOCITY ERROR (Vtrue-Vmig)

5

7

9

3

Dep

th (

km

)

4

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8

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BP MODEL DENSITY

5

7

9

3

Dep

th (

km

)

4

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RTM MIGRATION (Vmig) + IRREG. ACQUISITION

5

7

9

3

Dep

th (

km

)

4

6

8

10

OPTIMAL APERTURE (Vmig+IRREG. ACQUISITION)

5

7

9

3

Dep

th (

km

)

4

6

8

10

RTM MIGRATION (Vtrue+REGULAR ACQUISITION)

ZOOM: RTM MIGRATION (Vmig) +

IRREGULAR ACQUISITION

ZOOM: OPTIMAL APERTURE

(Vmig+IRREGULAR ACQUISITION)

5

7

9

3

De

pth

(k

m)

4

6

8

10

5

7

9

3

De

pth

(k

m)

4

6

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10

Raw aperture gathers

Optimized aperture gathers

• Common aperture CIGs (Kirchhoff type) available for band-limited

migration (common-shot migration)

• Common aperture CIGs can be used to produce superior quality

images

• Aperture comes almost for free if surface offset already computed

Conclusion

I would like to thank Total to give the permission to publish this

work

Acknowledgements

Common aperture plane x = -100 m

artifacts

Common aperture plane x = -800 m

Common aperture plane x = 2000 m Common aperture plane x = 500 m

RTM Common aperture (x) domain

S/N separation: imposing sparseness in the

solution

min f(m) f(m) = || Am – d||2 + l ||Wm||2

Preconditioned solution: P = W-1

min f(p) f(p) = || APp – d||2 + l ||p||2

PTP = Cm

Define image partitions as function of shot

position contained within the multi-fold

summation (VOO, VIP, etc.)

1 2

5 4 …

… … …

Aperture

S R

Dip vector

Common Shot aperture: definition #1

3

Surface acquisition binning

3

Define image partitions as function of shot

position contained within the multi-fold

summation (VOO, VIP, etc.)

1 2

5 4 …

… … …

Aperture

S R

Dip vector

• Proxy for both incident and scattered energy

• Need to compute aperture classes after migration

Common Shot aperture: definition #1

Surface acquisition binning

Aperture

S R

• Distance between extrema of active receiver position and those of

computation domain

• Equivalent conventional aperture varies significantly from trace to trace

Aperture

Common Shot aperture: definition #2